Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\frac{x^2+6xy+y^2}{x+y}$ on the blackboard. After $2023$ moves, only one number $c$ will remain on the blackboard. Prove that \[ c<2024 (a_1+a_2+\ldots+a_{2024}).\]